N-Particle Kinematics and Group-Theoretical Treatment of Phase Space I. Nonrelativistic

Abstract

This paper is a group-theoretical study of the kinematics of n nonrelativistic particles. A systematic method is given to construct a new complete set of commuting observables. The method is based on the existence of a group (the ``great group'') which acts transitively on the phase-space manifold and preserves the phase-space volume element; the observables are then Casimir operators of the great group and of some of its subgroups, including the usual three-dimensional rotation group. Among these collective observables, in addition to the total angular momentum, the most interesting is the ``togetherness operator'' which describes the simultaneous localization of the n particles. This operator is a generalization to n > 2 of the square of orbital angular momentum; its use allows to generalize to n particles the familiar centrifugal barrier arguments.